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Regarding AoPS Prealgebra, can I just say...


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It is SO DARN HARD I can hardly believe it! I feel like such an idiot, and I always thought I was fairly decent at math, went to an excellent private high school, took Calculus in college. There is no way I can figure out how to do some of these problems! I told ds today that we'll have to skip several problems and see if dad (engineer type) can explain them because I don't even understand the answer key. My son is hating it more by the day because it is so hard. He is a very sharp kid (scored 99 in ITBS math, top of his class at his private school) and is not used to working hard. I don't think he's REALLY hating it, and I'm hoping he'll come around. I did talk with him about it today and said I know it's hard - it's hard for me, but that's good for us. I told him we might have to refer to the answer manual more than we'd like to, but we're still learning that way and hopefully we'll get better as we go along.

 

But can I just also say that in spite of all this, or maybe because of it!, I think it is awesome! It is really make us look at math in a different way. It definitely feels more like something set up for participating in math competitions, BUT I hope that kind will serve him well in high school and beyond. We are having to look at the TM a lot, but I'm hoping we'll start to get the hang of it. I especially love how we're learning all sorts of cool shortcuts to hard problems. I don't really mean shortcuts, but being able to see how we might factor or distribute a complicated problem, or just subtract a 1 so we could then factor, etc.

 

For anyone who is considering it, I highly recommend it even though we're only on chapter 2! I do think it would probably be a bit much for kids that find math difficult to begin with.

 

Just wanted to share my thoughts. I'm thoroughly impressed and exasperated at the same time!

 

ETA: DS definitely isn't really hating it. Exaggeration. It's kind of a joke when we pull it out and we both laugh when he says, "I don't have to do AoPS today, do I?" and he does seem pretty satisfied (proud?) when he is able to work some of the problems. I shouldn't have said he was hating it more each day - that's more of a joke and of course people wouldn't realize that! If I told him he could quit, he would be thrilled, though!

Edited by HeidiKC
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Here's a hint, if you want one:

 

The lessons in PreA are set up pretty incrementally; one exercise in the first section builds up to the next, one step at a time, sometimes in pretty small steps.

 

Then you can use those steps later when you solve the exercises in the last part of the section, because they outlined the thought process for you. The writers didn't really want it to be a mystery on how to solve the problems-- even in the last exercise part of the section, earlier problems can often be used to solve later problems in the section.

 

That's why it's called a "discovery method" approach-- partly because they have you solve some problems before they present the material-- but also because one thing leads to another . . . but you are being led along a pretty definitively and narrowly defined path by the authors. If stuck, go back to previous problems and see if you can see anything you can use.

 

Or go back to the problems worked out in the initial section and see if there is something there you can use.

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Here's a hint, if you want one:

 

The lessons in PreA are set up pretty incrementally; one exercise in the first section builds up to the next, one step at a time, sometimes in pretty small steps.

 

Then you can use those steps later when you solve the exercises in the last part of the section, because they outlined the thought process for you. The writers didn't really want it to be a mystery on how to solve the problems-- even in the last exercise part of the section, earlier problems can often be used to solve later problems in the section.

 

That's why it's called a "discovery method" approach-- partly because they have you solve some problems before they present the material-- but also because one thing leads to another . . . but you are being led along a pretty definitively and narrowly defined path by the authors. If stuck, go back to previous problems and see if you can see anything you can use.

 

Or go back to the problems worked out in the initial section and see if there is something there you can use.

 

Thank you. You're right, but we're still finding it very challenging, mainly the challenge problems at the end of the chapter (the rest were manageable). This for instance:

 

How many digits are in the product of these two numbers:

9999....99 x 444...44 (where each number has 94 digits)

 

So they change 9999...99 to 1000...00 - 1 and then distribute, etc. It is modeled on some of the earlier problems, but it was even hard for me them to carry out the subtraction to get 4444...4435555...556. I'm leaving out info, but for you math heads, you probably get what I'm talking about. I think it turns out I've either gone downhill a lot in 30 years, or I just never had any really hard math. Probably some of both! But it is very hard for my son as well, maybe because he's used Saxon math up until now. I have to admit, he gets some of this AoPS faster than I do!

 

But I think it's a great book, a great approach and am excited to continue. And I am VERY thankful for the solutions manual! We're also using some other prealgebra materials. Something that is free and that I also really like is Lucid Math prealgebra videos. I'm having him watch those and take notes. They are short (3 min each) and very clear. Covers the basic concepts, nothing hard so far, but I think it'll be good to cement basic concepts in an easier way as well.

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I got it to self study in anticipation of my kids learning in the future and I feel the same. My first shock was the size of the book. My second shock was how little I apparently know.

The only positive surprise so far is when I handle those difficult problems well it's only because I have been working with Beast with my younger and got some training. I am hoping by the time my kids get through all the BA they plan to publish, their brain will handle this much better than my out of shape head.

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Just curious Heidi, what did your son use before this?

 

He used Saxon Math from K-6. And did very well. He also did LoF Fractions, and is almost done with Decimals and Percents. He's also done a lot of the Key to books: Fractions, Decimals, Percents, Measurements, Algebra (he's on book 3). But none of those have any hard problem solving. I'm sorry we waited so long for it!

 

I also had a small group of 4-8 kids this summer to meet with a math teacher for a little problem solving class. The kids were all incoming 7th graders, and my son was definitely ahead of most of them as far as problem-solving. But I think that's just because he's pretty quick and also good at math. Not because he knows how to really figure out hard stuff.

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Yes, but to balance that, it's a very algebraic pre-algebra. I won't feel bad if it takes us twice as long as normal pre-algebra would be (heck, with any other curriculum, dd would be taking algebra this year). Wonderful book!

 

Two tips: Take advantage of the online videos. And don't necessarily try to do a whole lesson---use a time limit, not a page or lesson goal.

 

Dd is only working a half hour a day, because that's all the hard thinking she can manage before her brain starts to mush out and the tears come. Life's been much better since I decided to let her stop before that point!

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I got it to self study in anticipation of my kids learning in the future and I feel the same. My first shock was the size of the book. My second shock was how little I apparently know.

The only positive surprise so far is when I handle those difficult problems well it's only because I have been working with Beast with my younger and got some training. I am hoping by the time my kids get through all the BA they plan to publish, their brain will handle this much better than my out of shape head.

 

That was my first reaction - so many words in a math book. I'm treating it as a read aloud, assigning a couple of problems, and then using one of the other math books.

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Thank you. You're right, but we're still finding it very challenging, mainly the challenge problems at the end of the chapter (the rest were manageable). This for instance:

 

How many digits are in the product of these two numbers:

9999....99 x 444...44 (where each number has 94 digits)

 

So they change 9999...99 to 1000...00 - 1 and then distribute, etc. It is modeled on some of the earlier problems, but it was even hard for me them to carry out the subtraction to get 4444...4435555...556.

Something I had to tell DS when he started with AoPS is that if they ask for the number of digits in the product (for instance) you don't need to work it out and do the subtraction. Just figure out enough to answer the digits questions. So in this case, the 94-digit 444...444 number times the 95-digit 1000...000 number will end up as 444...something with 94 more zeros on the end. Right? So 94+94 digits is 188 digits. Now subtracting *anything* from that is only going to affect the number of digits if it's close to the same size (at lest 187 digits itself, right?) and you know that 1*444...444 isn't, so skip that step. It's 188 digits.

 

I don't know if prealgebra gets to it or not (we started with a later book) but this is absolutely vital in the "what's the last digit" problems. You almost never need to work out the problem to figure out the last digit, and in fact it's usually meant to be nearly impossible to do it the long way. What you need to do is figure out which parts are required and which are extraneous.

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Something I had to tell DS when he started with AoPS is that if they ask for the number of digits in the product (for instance) you don't need to work it out and do the subtraction. Just figure out enough to answer the digits questions. So in this case, the 94-digit 444...444 number times the 95-digit 1000...000 number will end up as 444...something with 94 more zeros on the end. Right? So 94+94 digits is 188 digits. Now subtracting *anything* from that is only going to affect the number of digits if it's close to the same size (at lest 187 digits itself, right?) and you know that 1*444...444 isn't, so skip that step. It's 188 digits.

 

I don't know if prealgebra gets to it or not (we started with a later book) but this is absolutely vital in the "what's the last digit" problems. You almost never need to work out the problem to figure out the last digit, and in fact it's usually meant to be nearly impossible to do it the long way. What you need to do is figure out which parts are required and which are extraneous.

 

Thank you for this! I am going to print it out and try to figure out what you're talking about...cause I only kind of get it reading it now. But I think after a cup of coffee and a bit more focus it will make perfect sense. Because I do know that this problem (and so many others) aren't about doing a bunch of multiplying, etc., but instead it's about figuring out the method for this type of problem. And once you learn the secret, it's not so hard. I never remember doing any problems in my life where I had to figure out what's the last digit or how many digits, etc. Maybe I just have a bad memory, though. I do think this book is really going to each us a different way of looking at numbers and math problems.

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Thanks for sharing this. I am trying to decide which program to use for higher level math and someone suggested this in another thread. Do you think you will continue with it all the way through, up to Calculus?

 

And about the shortcuts you mentioned, that was one thing that stood out to me in the intro video about the VideoText Interactive program. The author said they don't teach any shortcuts and you don't need to memorize any formulas or shortcuts because they help you to understand the material, so memorizing a shortcut or a formula isn't necessary. What's your opinion about that so far? (I honestly know nothing about either program, I am just trying to compare them based on my very limited knowledge.) Thanks!

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. I am trying to decide which program to use for higher level math and someone suggested this in another thread. Do you think you will continue with it all the way through, up to Calculus?

 

 

 

I am not the poster to whom you have responded, but I am using AoPS through calculus. My DD is currently half way through the calculus book, and we have used most of the books (except for prealgebra which did not yet exist and number theory.)

 

And about the shortcuts you mentioned, that was one thing that stood out to me in the intro video about the VideoText Interactive program. The author said they don't teach any shortcuts and you don't need to memorize any formulas or shortcuts because they help you to understand the material, so memorizing a shortcut or a formula isn't necessary. What's your opinion about that so far?

 

Shortcuts are not necessarily tricks. And yes, I agree, they are not something to memorize. Conceptual understanding eliminates the need to memorize mathematical formulas almost entirely.

The types of shortcuts the pp mentioned arise from a thorough conceptual understanding; you will become more familiar with math and "see" where you can simplyfy your solution. AoPS is great at getting students to think before starting to calculuate. And teh cumulative effect of thinking about so many different problem develops the problem solving skills.

It is NOT really about finding the number of digits and training a shortcut for this type of problem, because it really is not a specific skill that has to be practiced. These problems are training tools to develop mathematical reasoning. You can do pretty well in math without knowing how to figure out the last digit of some number - but tworking this type of problems gets the student to think about math.

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Thank you! That was one point that I liked about the VideoText, in their intro they said that they teach you to understand it, and so you don't need to memorize a lot of formulas, because you will know it because you understand it. So would you sat AoPS is the same then in that regard? Do the videos online correspond well with working through the book? Because I feel like even I definitely need that person talking to help me understand so I can help him.

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Thank you! That was one point that I liked about the VideoText, in their intro they said that they teach you to understand it, and so you don't need to memorize a lot of formulas, because you will know it because you understand it. So would you sat AoPS is the same then in that regard?

 

 

Absolutely. AoPS teaches for conceptual understanding. If you understand, there is nothing to memorize- you can always derive what you need.

 

Do the videos online correspond well with working through the book? Because I feel like even I definitely need that person talking to help me understand so I can help him.

 

I have never used the videos; we work solely with the book. Most of the time, my kids don't even need our help; they just self teach from the books.

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AOPS has done amazing things for my DS. When he finished his lesson today, he danced around the table singing, "I am a math genius!". It has boosted his self confidence in a way no other curriculum ever has. He is extremely visual-spatial and dyslexic, so he has always struggled with every other curriculum. I am so happy we discovered it!

For anyone having problems, definitely watch the video lesson first, then make sure your child understands the property outlined in each lesson before attempting any problems. It took my DS a few lessons to catch on, because it is so unlike anything he has done before. After a short adjustment period, he loves it. :001_smile:

I can't wait for youngest DD to get old enough to use Beast Academy. :D

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Okay, I guess I'm going to have to wait for the book to see this, but how can he do the problems before watching the video? And how are exercises different from problems? No need to answer me, I am sold on getting it, so I will wait and see! Thanks for this discussion. :)

 

I'll attempt to answer in case anyone else is curious. :)

 

If you watch the videos first, the discovery aspect is reduced. The lesson problems are when the discovery takes place, so waiting to watch the videos allows more discovery. The exercise problems use what has been discovered.

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I am not the poster to whom you have responded, but I am using AoPS through calculus. My DD is currently half way through the calculus book, and we have used most of the books (except for prealgebra which did not yet exist and number theory.)

 

 

 

Shortcuts are not necessarily tricks. And yes, I agree, they are not something to memorize. Conceptual understanding eliminates the need to memorize mathematical formulas almost entirely.

The types of shortcuts the pp mentioned arise from a thorough conceptual understanding; you will become more familiar with math and "see" where you can simplyfy your solution. AoPS is great at getting students to think before starting to calculuate. And teh cumulative effect of thinking about so many different problem develops the problem solving skills.

It is NOT really about finding the number of digits and training a shortcut for this type of problem, because it really is not a specific skill that has to be practiced. These problems are training tools to develop mathematical reasoning. You can do pretty well in math without knowing how to figure out the last digit of some number - but tworking this type of problems gets the student to think about math.

 

:iagree: Great answer!

 

It isn't really about "shortcuts" to be memorized, it is about simplifying the solution. Thinking about the problem in a different sort of way, and finding a most lovely and simple way to work it out! We're only on chapter 2, but I keep telling DS that if he's doing all sorts of complicated figuring (multiplying large numbers), it's likely he's making it harder than it needs to be and that he should consider if there is a simpler way to do it - a "shortcut", I guess you could call it. Nothing he's memorized. But look at the problem closely and see if there is something you can factor out, or maybe do something else with that exponent rather than multiplying the number out against itself 21 times, etc. We're slowly getting the hang of it.

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Thanks for sharing this. I am trying to decide which program to use for higher level math and someone suggested this in another thread. Do you think you will continue with it all the way through, up to Calculus?

 

He'll be going to a private high school, but I would not likely choose this to use through Calculus if I was going to continue homeschooling him. If we both completely got it, maybe. But I'm thinking that as much as we're struggling with pre-algebra, it would be pretty scary to continue through Calculus. I feel much better now that he is also using Life of Fred Pre-algebra and also Key to Algebra. I don't know that I'd like using just AoPs pre-algebra without a more traditional curriculum alongside (not that LoF is particularly traditional!). It's just reassuring to me that he's using something else that he completely understands.

 

I think I might possibly continue with the series as a supplement, because it really is so good. And it really makes you think differently. My husband cracked it open for the first time tonight to help with a few problems. He didn't think it was super hard! But then he has an Electrical Engineering degree. He liked the book and said something like, "You just have to look at the problem closely to figure out what it is they want you to do".

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AOPS has done amazing things for my DS. When he finished his lesson today, he danced around the table singing, "I am a math genius!". It has boosted his self confidence in a way no other curriculum ever has. He is extremely visual-spatial and dyslexic, so he has always struggled with every other curriculum. I am so happy we discovered it!

For anyone having problems, definitely watch the video lesson first, then make sure your child understands the property outlined in each lesson before attempting any problems. It took my DS a few lessons to catch on, because it is so unlike anything he has done before. After a short adjustment period, he loves it. :001_smile:

I can't wait for youngest DD to get old enough to use Beast Academy. :D

 

Yes! DS complains all over the place when I say it is time for AoPS. But everything always comes so easily for him, and this is honestly probably the first time he's struggled with a course - and understandably does not enjoy that feeling! But when he is successful, he feels so good about himself. I love that. So what if he gets 100% on something that is easy for him. I think we'd both be thrilled and proud if he got an 85% on an AoPS test!

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:iagree: Great answer!

 

It isn't really about "shortcuts" to be memorized, it is about simplifying the solution. Thinking about the problem in a different sort of way, and finding a most lovely and simple way to work it out! We're only on chapter 2, but I keep telling DS that if he's doing all sorts of complicated figuring (multiplying large numbers), it's likely he's making it harder than it needs to be and that he should consider if there is a simpler way to do it - a "shortcut", I guess you could call it. Nothing he's memorized. But look at the problem closely and see if there is something you can factor out, or maybe do something else with that exponent rather than multiplying the number out against itself 21 times, etc. We're slowly getting the hang of it.

 

My ds does the same thing - he totally overcomplicates the questions. It usually takes just a few questions from me before he "sees" the answer and then he is shocked by how easy it actually was. He's the kind of guy who can't see the forest through the trees, but I am hoping that he starts looking at the bigger picture.

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Aops is proving to be a great tool to help my child focus on "the big picture" as opposed to the details. Now, he still needs to nail those details, but I expect him to nail them...they are not the centerpiece anymore of math. Aops have moved the center from practice, learning "how" to do things to the far more interesting question of Why? The transition from one perspective to another is challenging, but i think he is beginning to see the light.

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Aops is proving to be a great tool to help my child focus on "the big picture" as opposed to the details. Now, he still needs to nail those details, but I expect him to nail them...they are not the centerpiece anymore of math. Aops have moved the center from practice, learning "how" to do things to the far more interesting question of Why? The transition from one perspective to another is challenging, but i think he is beginning to see the light.

 

This gives me hope. I have ordered AoPS pre-algebra and my ds is looking forward to it, though I have warned him that it will be unlike anything he's done before and will likely be very challenging. He seems okay with that. He loves to know why. why, why, why, why why? Perfect. Let's hope this works out. I'll admit that I'm a little nervous, as I am not a math whiz at all.

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So but this makes me nervous. Does the student need someone who can ask those questions to be able to discern that? Or do they learn that from the curriculum itself? Because if he needs me to ask the right questions to draw that out of him, I'm not sure I can do that.

 

 

 

It isn't really about "shortcuts" to be memorized, it is about simplifying the solution. Thinking about the problem in a different sort of way, and finding a most lovely and simple way to work it out! We're only on chapter 2, but I keep telling DS that if he's doing all sorts of complicated figuring (multiplying large numbers), it's likely he's making it harder than it needs to be and that he should consider if there is a simpler way to do it - a "shortcut", I guess you could call it. Nothing he's memorized. But look at the problem closely and see if there is something you can factor out, or maybe do something else with that exponent rather than multiplying the number out against itself 21 times, etc. We're slowly getting the hang of it.

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Does the student need someone who can ask those questions to be able to discern that? Or do they learn that from the curriculum itself?

 

My boys enjoyed the "shortcuts" from the AOPS pre-algebra videos so much that I bought the pre-algebra book for them. They can imitate Richard Rusczykvery well.

 

My older boy is finishing singapore math 6 and doing topics that he like from the pre-algebra book. I have not need to buy the solutions manual yet. We will be doing the pre-algebra book systematically when he finish singapore math 6.

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So but this makes me nervous. Does the student need someone who can ask those questions to be able to discern that? Or do they learn that from the curriculum itself? Because if he needs me to ask the right questions to draw that out of him, I'm not sure I can do that.

 

The books are written TO the student and do not require a teacher. The discovery of the concepts does not happen in a vacuum; the student is guided to make the discovery by carefully designed problems that lead him there.

The curriculum teaches problem solving through critical thinking; the most important thing is that the student develops the mindset that math is something to think about, not something where you drill or press buttons. And AoPS does a really good job developing this.

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So but this makes me nervous. Does the student need someone who can ask those questions to be able to discern that? Or do they learn that from the curriculum itself? Because if he needs me to ask the right questions to draw that out of him, I'm not sure I can do that.

 

There are hints for the particularly difficult problems. You'll see a hint number next to it and can turn to the back of the book and find that hint number. The "problems" are fully worked out and fully explained (that's the teaching - all teaching is in the text, written to the student). The solutions manual has all problems worked out fully so it is obvious which concept you apply to the problem.

 

AoPS is designed to be self-teaching.

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The "problems" are fully worked out and fully explained (that's the teaching - all teaching is in the text, written to the student).

 

:iagree:

Which is why it is very important to read their solutions carefully, even if the student was able to solve a problem independently: the solution discusses the concept, and sometimes the solution may differ from the one the student found (often there are several different correct ways to solve one problem).

It also explains why some people consider AoPS "wordy": because everything a live teacher would say when teaching the concept is right there in the text.

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:iagree:

Which is why it is very important to read their solutions carefully, even if the student was able to solve a problem independently: the solution discusses the concept, and sometimes the solution may differ from the one the student found (often there are several different correct ways to solve one problem).

It also explains why some people consider AoPS "wordy": because everything a live teacher would say when teaching the concept is right there in the text.

This is one of the things I love about AOPS. They will show the student different ways to get the answer, even show common mistakes. I am going through the Algebra book alongside my son and am amazed at what I am learning. I thought I knew Algebra but was sorely mistaken. This is by far the best math curriculum I have found. They say it is for advanced math students but I think it can be used by even those who detest math(my son). He is stretched, yes, but he is learning because he is self discovering. Just take it slow.

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Future AoPS lovers here; I say future because the we're still on a big learning curve. Oldest DD, 7th grade is working through Chapter 1 at the moment and I have a concern. She's working through the text on her own, which she likes doing. However, she gets many/most of the problems wrong. I'm not concerned about her math ability, but I think she could have a bit more "grit". How should I respond? If we go through the solutions together, I think the discovery method is compromised. I'm guessing I should just send her back to the explanation and try again. How many times would you do that before giving assistance?

 

Thanks for your insight!

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Future AoPS lovers here; I say future because the we're still on a big learning curve. Oldest DD, 7th grade is working through Chapter 1 at the moment and I have a concern. She's working through the text on her own, which she likes doing. However, she gets many/most of the problems wrong. I'm not concerned about her math ability, but I think she could have a bit more "grit". How should I respond? If we go through the solutions together, I think the discovery method is compromised. I'm guessing I should just send her back to the explanation and try again. How many times would you do that before giving assistance?

 

Thanks for your insight!

 

 

I am by no means an expert as we have only been using AoPS for 2 weeks. However, I think the first chapter can be seen as a transition chapter from the "old way" she was doing math, which was more straightforward probably, to the "new way". For the first chapter, perhaps have her try the problems, watch the videos, and then go back and try again?

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For our first two chapters of Prealgebra, we basically went through it all together. I started to "detach" in chapter three. It takes a while to understand that it's ok to try a problem you haven't been taught how to do.

I'm really bad at analogies, but it's like you are traveling and have reached a river. With old math, they would say, here's the river, and there is the boat you use to cross it. With AoPS, they would say, here's the river, and you have to figure out how to cross it. And so, you think about it for a while, try a few things, and end up with a boat of your own. It may not look anything like the other person's boat, but it gets you across. It was super hard to do, but hey, now you know that you CAN make your own boat, and next time, it will be easier. And maybe your boat will be more elegant next time. The transition period is when you are helping the other person to build a boat, since they've never done it before.

 

Ok, so I got a little carried away... :D

In my boat.

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The books are written TO the student and do not require a teacher. The discovery of the concepts does not happen in a vacuum; the student is guided to make the discovery by carefully designed problems that lead him there.

The curriculum teaches problem solving through critical thinking; the most important thing is that the student develops the mindset that math is something to think about, not something where you drill or press buttons. And AoPS does a really good job developing this.

 

:iagree: Although I do think it would be nearly impossible for my son to do this without some help from me (and I need the solutions manual!). But I do see him improving as we move further into the book - getting the hang of this new approach. Even though the concepts are explained in the problems section, sometimes there are exercises that he can't quite figure the approach for, or quite how to work it. I sometimes have to give him hints. He has always done his math completely on his own and never needed help. This is the first time I've ever had to help him with math. Which is fine. He's learning a lot, even this way.

 

I do have a question: is there anyone out there who has a child who is a pretty average math student who is able to understand and use the curriculum easily, and without much help from you? And if so, what math curriculum did you use previously?

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Future AoPS lovers here; I say future because the we're still on a big learning curve. Oldest DD, 7th grade is working through Chapter 1 at the moment and I have a concern. She's working through the text on her own, which she likes doing. However, she gets many/most of the problems wrong. I'm not concerned about her math ability, but I think she could have a bit more "grit". How should I respond? If we go through the solutions together, I think the discovery method is compromised. I'm guessing I should just send her back to the explanation and try again. How many times would you do that before giving assistance?

 

Thanks for your insight!

 

We did chapter 1, and the boys thought they had it nailed. Then we started chapter 2 and they realized that they hadn't really understood the implications of chapter 1. So they re-did that chapter. It was a strong lesson into the need to actually work through the sample problems themselves, not just review the solutions.

 

ETA: This was for the algebra book. We didn't do pre-algebra.

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  • 11 months later...

I am not the poster to whom you have responded, but I am using AoPS through calculus. My DD is currently half way through the calculus book, and we have used most of the books (except for prealgebra which did not yet exist and number theory.)

 

 

 

Shortcuts are not necessarily tricks. And yes, I agree, they are not something to memorize. Conceptual understanding eliminates the need to memorize mathematical formulas almost entirely.

The types of shortcuts the pp mentioned arise from a thorough conceptual understanding; you will become more familiar with math and "see" where you can simplyfy your solution. AoPS is great at getting students to think before starting to calculuate. And teh cumulative effect of thinking about so many different problem develops the problem solving skills.

It is NOT really about finding the number of digits and training a shortcut for this type of problem, because it really is not a specific skill that has to be practiced. These problems are training tools to develop mathematical reasoning. You can do pretty well in math without knowing how to figure out the last digit of some number - but tworking this type of problems gets the student to think about math.

 

Wish there was a double like for this answer!

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